| L(s) = 1 | + 1.75·2-s + 3-s + 1.08·4-s − 2.67·5-s + 1.75·6-s + 0.0258·7-s − 1.60·8-s + 9-s − 4.70·10-s + 1.36·11-s + 1.08·12-s − 2.10·13-s + 0.0453·14-s − 2.67·15-s − 4.99·16-s − 2.63·17-s + 1.75·18-s − 5.19·19-s − 2.91·20-s + 0.0258·21-s + 2.39·22-s − 23-s − 1.60·24-s + 2.16·25-s − 3.70·26-s + 27-s + 0.0280·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.577·3-s + 0.544·4-s − 1.19·5-s + 0.717·6-s + 0.00975·7-s − 0.566·8-s + 0.333·9-s − 1.48·10-s + 0.410·11-s + 0.314·12-s − 0.583·13-s + 0.0121·14-s − 0.690·15-s − 1.24·16-s − 0.639·17-s + 0.414·18-s − 1.19·19-s − 0.651·20-s + 0.00563·21-s + 0.510·22-s − 0.208·23-s − 0.326·24-s + 0.432·25-s − 0.725·26-s + 0.192·27-s + 0.00531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 0.0258T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 + 9.24T + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 + 0.699T + 47T^{2} \) |
| 53 | \( 1 + 0.391T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 3.62T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 + 9.67T + 79T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 3.61T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.476048602235439890527103445130, −8.172196551381714719233125524153, −6.87043536491638651355790472496, −6.57267979726791952681872614105, −5.20339725897035646235511889611, −4.50030500749884654998510872217, −3.87046681834299117226543085567, −3.16731961325277590139019571727, −2.08664394595949765359949195160, 0,
2.08664394595949765359949195160, 3.16731961325277590139019571727, 3.87046681834299117226543085567, 4.50030500749884654998510872217, 5.20339725897035646235511889611, 6.57267979726791952681872614105, 6.87043536491638651355790472496, 8.172196551381714719233125524153, 8.476048602235439890527103445130
